Borel was a great mathematician who did work in measure theory and probability theory. He originated the idea of an infinite number of monkeys typing. This is the basis for physical infinity providing whatever conditions a mathematician desires somewhere in the universe. Although Borel's image was to show that such results were more likely than a gas departing significantly from thermodynamic equilibrium, as imagined by Ludwig Boltzmann and his critics..

Mathematical infinity is the basis for philosopher David Lewis's idea that any "possible" condition (indeed a whole universe) is "actual" (realized) somewhere. Cosmologist David Layzer has a similar theory for multiple realizations of any possible condition in the single infinite universe. And Arthur Stanley Eddington described perfect repetitions of his Messenger lectures given an infinite time.

Interventionism

Borel wrote an influential book on Chance (Le Hasard), which appeared in four editions between 1914 and 1948. Unlike many mathematicians, he argued that determinism was a "pure abstract fiction," because of the "indetermination" of the initial data. Borel made an oft-quoted calculation that a single gram of matter moved one centimeter in a distant star would be enough to randomize the motions in a terrestrial gas. This is now known as interventionism. Because no system is completely isolatable, the argument is that external interventions destroy the molecular correlations that are necessary for reversibility.

The representation of gaseous matter by a model, composed of molecules with positions and velocities which are rigorously determined at a given instant is therefore a pure abstract fiction... as soon as one supposes the indeterminacy of the external forces, the effect of collisions will very rapidly disperse the trajectory bundles which are supposed to be infinitely narrow, and the problem of the subsequent movement of the molecules becomes, within a few seconds, very indeterminate, in the sense that a colossal number of different possibilities are a priori equally probable.

(Le Hasard, Second edition, 1924.)

This claim is absurd that a miniscule conservative gravitational force coming from outside the theoretically isolated container of gas will change the evolution of the gas from completely deterministic and reversible to indeterministic and irreversible within a few seconds.

First, the gravitational force of 1 gm of matter at Sirius' distance of 8.6 light years is 10-12 of the force of just one other atom in a 1 liter volume of gas. Second, the angular rate of change of the nearby atom is of the order of 1 radian in 10-8 seconds, whereas the 1 cm/sec at the Sirius distance is only 10-18 radians per second, so the differential force is of order 10-26 weaker.

But third, and even more important, as long as the Hamiltonian including the remote gram of matter is still symmetric under time reversal, including the distant object makes absolutely no difference in the statistical mechanics.

Borel's idea has been used to claim that Joseph Loschmidt's reversibility objection to Boltzmann's H-Theorem could be easily invalidated. Of course, since the distant gram of matter moves in the opposite direction under time reversal - Borel's suggestion makes no difference.

In his 1964 book, Scientific Uncertainty and Information, Leon Brillouin cited Borel's 1914 Introduction géométrique a quelques théories physiques (p.94) as explaining how an external disturbance could randomize the motions of molecules in a terrestrial gas.

C. It is impossible to study the properties of a single (mathematical) trajectory. The physicist knows only bundles of trajectories, corresponding to slightly different initial conditions.

Note that it is Brillouin, not Borel, who suggests Sirius

Borel, for instance, computed that a displacement of 1 cm, on a mass of 1 gram, located somewhere in a not too distant star (say, Sirius) would change the gravitational field on the earth by a fraction 10-100. The present author went further and proved that any information obtained from an experiment must be paid for by a corresponding increase of entropy in the measuring device: infinite accuracy would cost an infinite amount of entropy increase and require infinite energy! This is absolutely unthinkable.

D. Let us simplify the problem, and assume that the laws of mechanics are rigorous, while experimental errors appear only in the determination of initial conditions. ln the bundle of trajectories defined by these conditions, some may be "nondegenerate" while others may "degenerate." The bundle may soon explode, be divided into a variety of smaller bundles forging ahead in different directions. This is the case for a model corresponding to the kinetic theory of gases. Borel computes that errors of 10-100 on initial conditions will enable one to predict molecular collisions for a split second and no more. It is not only "very difficult," but actually impossible to predict exactly the future behavior of such a model. The present considerations lead directly to Boltzmann's statistical mechanics and the so-called "ergodic" theorem.

(Scientific Uncertainty, and Information, p.125)

It is clear that some authors quoting Borel (e.g., David Layzer, H. Dieter Zeh, Pine and Golub) with Sirius as the intervening star have not really read Borel. They have read Brillouin, without always citing him directly. Brillouin did not invent Sirius. Borel actually mentions Sirius, but not in the context of his "interventionist" explanation for the resolution of the Loschmidt reversibility objection and argument against the "fiction" of deterministic.

We seek to evaluate, starting from Newton's law of gravitation, the deviation of a molecule, between two collisions, caused by the displacement of a mass extremely small located at a great distance. Under the action of gravity a heavy body falls in one second five meters, in 10 -10 seconds, it will fall by an amount 10 -20 times smaller; if, instead of the earth, we consider a small concentric sphere with the same density and whose linear dimensions are 10 17 times lower (the circumference of a great circle is 4 ten -millionths of a millimeter instead of 40 million meters), the mass of the sphere being 10 51 times lower than that of the earth, the deviation would be even smaller, 10 20 X 10 51 = 10 71 times smaller. Now this tiny ship in the sphere beyond the ends of the visible universe, at a point where light takes billions of years to reach us, instead she would one-fiftieth of a second to come from the center of the earth, the distance being about 10 18 times bigger, the attraction becomes 10 36 times lower, the final deviation is 10 26 X 71 X 10 = 10 107 times lower. Finally, in a similar calculation that I omit, the value of the deviation would correspond, either to the action of the tiny sphere placed at the prodigious distance, or the simple effect of moving a millionth of millimeter in the position of such a sphere that far.

Even in accumulating the assumptions likely to make the action as low as possible, we can not even divide the deviation by 10200, that is to say that the changes produced by such an action in the path of a molecule are colossally large compared to what has been necessary for our argument: The representation of a gaseous mass with a single model, consisting of molecules whose positions and velocities at a given time are rigorously determined, is pure abstract fiction; it can not be closer to reality than imagining a stack of models, that is to say, by assigning to the initial data some uncertainty. So weak as this uncertainty, however small one supposes the uncertainty of external forces, the effect of collisions quickly disperses the beams trajectories assumed infinitely thin strokes and the problem of subsequent movement of molecules is in a very short of seconds, very indeterminate, in that a huge number of different possibilities are great a priori equally probable. Of course, in a moment, today, only one of these possibilities is realized, but the uncertainty is reborn as great as soon as it is the problem of the state at a future time, even very close. The only form in which the problem can be posed and solved is the statistical form: the large number of possible contingencies can be separated. into two very unequal groups, all those gathered in the largest group with certain common characteristics? Thus if we consider all possible contingencies for the state of a billion parts of heads or tails, it can be a first group with the case where the deveiationis less than 1,000,000, where the number of games won is between 499,000,000 and 501,000,000, a second group with all other cases. The first group is extremely larger than the first, it is highly probable that the event will be achieved is up to the first group, that is to say that this character will have the ratio of the number of parts battery parts face will be between 0.996 and 1.004. The conclusions reached by the application of statistical methods to the study of problems of the kinetic theory are similar in nature to the previous conclusion, and indeed because of the large number of molecules, they are even more precise about the meaning to be attributed to the highly probable words, we can only refer to the comparison of the miracle of monkeys typing.

68. - When looking, as we have been, at the problems of statistical mechanics, the objection called the Loschmidt objection can easily be lifted. This objection is the following. The application of kinetic theory to the study of thermodynamic phenomena led to the account of irreversible phenomena, such as the establishment of temperature equilibrium between two bodies in contact. However, the kinetic theory uses mechanical phenomena that are all reversible, that is to say that the equations that represent these phenomena are not modified when changing the sign of the time. To be more concrete, if we think, at a given instant, that we could reverse each molecule by giving it a velocity exactly opposite to its current speed, everything happens as if we are projecting a film in reverse, starting with the most recent portions. It is thus not possible, Loschmidt objected, to explain irreversible phenomena by means of a reversible mechanism. This objection falls when one has fully realized the necessarily statistical mechanical explanations; there is no attempt to determine rigourously strictly defined molecular mechanical phenomena, but to consider the most likely among all possible motions; this indeterminacy of the future is the very principle of statistical mechanics; but we can not speak of indeterminacy of the past, so the distinction between the past and the future, that is to say, the asymmetry of Carnot's principle regarding the sign of the time, does not contradict a mechanical explanation of the facts of thermodynamics. We will return later (ch. X) to the discussion of the theory of irreversibility.

I would like to say a few words about Boltzmann's remarks on the implementation of the second law of the universe. As Boltzmann rightly said, "surely no one will speculate about such important discoveries, nor about the highest goal of science, as did the ancient philosophers. But it is not certain that it would be just to turn in derision and to look at them as useless." Boltzmann developed a mechanical conception of the universe, in which there occurs here and there, passages from a state more probable to a state less probable, so that, for the whole universe, irreversibility does not exist. This conception is rigorous in the abstract if the universe is a mechanical system which can be defined by a finite number of parameters, and the total field of variation is finite. Assume for a moment, we can accept this image for the world we see, that is to say that we can fix a very large number R, such that there is nothing outside the sphere S of radius R, the sphere S is our universe; the evolution of this universe is, according to a theorem of Poincaré, as close as we want to a periodic evolution and in immensely long periods, phenomena contradict the second law will be as common as the phenomena in accordance with this law. Even leaving aside the difficulties - but I believe insurmountable - implied by the assumption that nothing leaves the sphere S, it must be noted that the conclusion is rigorous only so long as we assume the absolute absence of any external action on S. Imagine a sphere S2, whose dimensions relative to S are like those of S relative to an atom, and then a sphere S3 that would be to S2, as S2 is to S, and so on until a sphere Sn with the index n equal to one million. In order for the application to S of Poincaré's quasi-periodic mechanical theory to be legitimate, it would be necesary that there is not, within the sphere Sn, some universe S' of the same size as S and likely very different from S which can, in the course of time, act on S. For the length of time required for the application Poincaré's theorem is so long that a meeting of S with S' would be highly probable, well before this time had elapsed. That is to say that it is at least as plausible to assume that the laws of our universe will be completely changed by a combination with another universe (now much farther from ours than an atom located on the Earth is far from an atom located on Sirius) to assume that a significant change of direction in the change of entropy.

Borel knows that no system is ever completely isolated from the rest of the universe

In other words, I think that the regular evolution towards states more and more probable must be allowed for the entire universe, contrary to the concept of Boltzmann, as long as we do not look like a finite system isolated forever in a finite portion of space from which nothing can escape, neither matter nor energy or radiation and into which nothing can enter.

Conceive that we have prepared a million monkeys randomly hitting keys on a typewriter and that, under the supervision of illiterate foremen, these monkeys work hard typing ten hours a day with a million typewriters of various types. Illiterate foremen would gather the typed blackened and link them in volumes. After a year, these volumes contain an exact copy from books of all kinds and of all languages ​​stored in the richest libraries in the world. This is the probability that occurs during a very short time, in the container A, a deviation of about one hundred thousandth of the composition of the gas mixture. Assume that the deviation thus produced will remain for a few seconds is an admission that, for many years, our army of monkeys typing, working always in the same way every day, will supply an exact copy of all printed books and newspapers, which appear on corresponding day of the following week over the entire surface of the earth, including all the words to be uttered by all men on that day. It is easier to say that these differences are unlikely, purely impossible.